The manufacturing of integrated circuits involves the design and use of lithographic processes to transfer patterns representing integrated circuit layouts to a semiconductor substrate or wafer. The lithographic pattern transfer process includes the use of a photomask to transfer patterns on the photomask to a film stack on a semiconductor wafer. The pattern transfer process from a photomask to a film stack on a semiconductor wafer is routinely simulated during design and manufacture of the photomask using a variety of process models. One important application of this simulation is Optical Proximity Correction (OPC), where nonlinearities in the patterning process are corrected for by predistorting the mask in a manner so that the final pattern looks very similar to the desired target pattern. The process models used for this type of simulation are often divided into at least two components. The first component, the “optical model,” describing the optics of the lithography tool and typically simulates an optical image of the mask at some plane inside the photoresist material. The second component, the “photoresist model,” describes how the photoresist material responds to that optical image (exposure) and the subsequent processing of the resist (development) that is done to create a pattern on the wafer. So, the combination of an optical and photoresist model can be used to simulate how a photomask pattern will be transferred to a pattern on a wafer. Other pattern transferring processes, such as etch, can also be modeled in a similar fashion, for example, as a polynomial or other empirically calibrated function of image parameters, as known in the art.
Photoresist models are typically approximations to the true physical behavior and are used to describe the simulated location of a patterned edge. These approximations are often based on a model form made up of a polynomial with terms based on parameters of the optical image that exposes the resist. For example, this polynomial may have terms containing the maximum image intensity or the slope of the image intensity. In addition to linear terms in these parameters, there can also be higher order terms including quadratic terms in any of the image parameters and cross-terms between image parameters. In these types of models, each term in the polynomial has a corresponding multiplicative coefficient that is typically calibrated to experimental data using a least-squares fitting algorithm, with the result often being considered the “best-fit polynomial.” The best fit model is one which has coefficients which, given particular values of the independent variables, will most closely fit the emprically measured data according to some metric of goodness, such as a root mean square (RMS) error, even though the model may not exactly reproduce the calibration data. An example of such a polynomial is given in equation (1). Here, the photoresist response is given by R, the multiplicative coefficients are specified as an, and the terms I_max and slope represent the maximum image intensity and slope of the image intensity, respectively. Depending on the type of modeling being done, the photoresist response function R may represent different things. In some cases, it represents a threshold value that is applied to the optical image to determine where printed edges occur and, in other cases, it represents a latent image that is exposed in the photoresist, where printed edges occur when this image crosses a predetermined threshold. It is evident that linear terms (a1*I_max), quadratic terms (a3*slope2) and cross-terms (a4*I_max*slope) are used:R=ao+a1*I_max+a2*slope+a3*slope2+a4*I_max*slope  (1)
Best-fit polynomial models have been shown to very accurately represent the data that was used in the calibration of the coefficients and, when calibrated properly, to remain very accurate for data points interpolated over small distances between calibration points. FIG. 1 illustrates a plot of edge placement error (EPE) of simulated images versus measured edge placements for verification test patterns that were not included in the model calibration. FIG. 1 shows that the edge placements for patterns specified by the circle 101 are very accurately modeled by the model used in the simulations since their EPE values are close to zero. Due to the non-physical nature of the polynomial representation, however, these models do not typically have very good accuracy for data points that fall outside of the calibration set. These points require the model to extrapolate to image parameters that are beyond the values used in the calibration set. Accuracy may also suffer if the image parameters fall in a region where the model is interpolating between calibration points, but the distance from the nearest calibration point to the point of interest is large. Poor accuracy is seen for patterns 102 in FIG. 1 where the EPE values are relatively large negative values.
FIG. 2 illustrates three image parameter values, Imin, Imax and curvature, computed at selected evaluation points of a calibration test structure (hereinafter, “calibration points”). The optical parameters obtained at the selected calibration points, 203, are plotted. In addition, image parameters 202 are plotted for evaluation points on verification patterns that were determined to have poor accuracy. The plot shows that the verification image parameters 202, fell within a large gap in the calibration data 203. However, the image parameters for the verification pattern 202 still fell within the bounds defined by the largest and smallest calibrated values for each image parameter. These bounds are shown as rectangular projections, 201, onto each 2-dimensional plane in this 3-dimensional plot.
Controls must be put in place to ensure that a resist (or other polynomial fitted) model remains accurate under any arbitrary imaging conditions. This is done typically during calibration of the resist model 316 as illustrated in FIG. 3. A set of calibration patterns 302 are used to obtain measured image data (calibration data 301). The calibration patterns 302 are provided as input into the optical model 303 to obtain simulated images 304. The simulated images 304 are used to obtain the polynomial coefficients, and thereby calibrate the (Block 305) photoresist model 307, by fitting the polynomial to the measured image data 301 from calibration patterns 302. The simulated images 304 are also used to determine extreme values for each image parameter (306) for all of the calibration data points and these extreme calibration data points are stored with the stored model 318 as one dimensional (1D) model bounds 308. These one-dimensional model bounds 308 are equivalent to those shown in FIG. 2 (201) and are the only values that are stored to provide information on the expected accuracy of the photoresist model 307 for any arbitrary image.
These bounds 308 are normally used in one of two ways when modeling a lithographic process for a pattern layout 309 to simulate a photoresist image 315. The first is shown in block 317, where the optical model 310 is used to obtain the simulated image 311 transmitted from the optical system. The simulated image parameters 311 can be checked against the model bounds 308 and, if they are outside of the model bounds, the image parameters 311 used in the resist model 307 that would exceed the range of calibration data can be limited (i.e. reset) (Block 312) to be equal to exactly the model bound value 308 (Block 313). In this way, the image parameters 311 are never allowed to go out of bounds, so the resist model 307 should never be predicting the response of the resist (as computed in Block 314) in an optical regime beyond which the resist model 307 was calibrated. This is only the case, however, if every single term in the polynomial, including quadratic and cross-terms, has set bounds that are enforced. Often, a simplifying assumption is made that only the linear values of each image parameter need to be bounded. When limiting the image parameters 311 to the bound values 308, the assumption is made that the resist model 307 will be more accurate by using the “wrong” artificially set image parameters (i.e. the model bound values 308) than it will by using the “right” model predicted image parameters (i.e. the actually simulated image parameters 311 computed by the optical model) when the simulated image parameters 311 fall outside the model bounds 308. This approximation leads to significant uncertainty in the modeled response of the resist for regions when the image parameters 311 are limited to the model bounds 308.
The second way the bounds 308 are used is illustrated in Block 417 of FIG. 4, which is to just highlight regions where the image parameters 311 fall outside the model bounds 308. For example, in FIG. 4, the design layout 309 is used to obtain the simulated image 311 from the optical model 310, as in FIG. 3. However, in this example, the photoresist model 307 is used to compute the resist response 415 to the simulated images 311, but if the images are outside the bounds 308 of the calibrated data (Block 420), then these resist responses corresponding to the out of bounds simulated image parameters are tagged (Block 421). In this way, these regions of the resist image 415 can be shown to have questionable model accuracy. Again in this case, only the one-dimensional model bounds 308 have been used to highlight regions of questionable model accuracy. In this case, the resist model 307 is allowed to use the out-of-bounds image parameters as computed by the optical model (Block 310) and return a modeled resist response 414. However, the modeled resist image 415 may have portions marked as having large uncertainty. Such tags merely highlight regions that are “uncertain” without providing a quantitative measure of the uncertainty. In addition, the cutoff between “certain” and “uncertain” is based solely on the one-dimensional model bounds 308.
In view of the foregoing, it would be desirable to provide a method to quantify the uncertainty of resist or other polynomial models of image parameters.